Can an under-target and under-volume wort be fixed by adding more wort of higher gravity?

Question

Here's the situation: today I brewed a Kölsch style ale, and I encountered two problems:

1. The mash wasn't very efficient: 60% instead of 70%. OG 1.038 instead of 1.045.
2. It came in under-volume: 4.5gal instead of 5.5gal. I forgot to factor loss from the hops soaking up wort.

My plan to fix both problems is that tomorrow I will brew 1gal of 1.076 wort and add it to the fermenter to gain back both volume and gravity.

Questions:

1. Does this plan sound reasonable? Any drawbacks or pitfalls you can think of? Have I done the math right?
2. Do I need to include hops in the new wort addition? I am unsure if the hop utilization of the original wort is sufficient for the full 5.5gal volume or if I need to factor hops into the addition. The wort as-is tastes a little on the bitter side for a Kôlsch, but I've never done a Kölsch before so maybe that's normal at this stage.
3. Am I over thinking this? Is there an easier way?

Worth noting that I haven't yet pitched the yeast. The wort and the 1q yeast starter (Wyeast 2565 Kölsch; 1/2 cup pils DME; 48 hours on it) are sitting in the ferm chamber at 60F. I assume that no significant problems will be created by letting the wort sit overnight.

The recipe is pretty simple: 8lb pils, 1lb Vienna, .5lb wheat malt; mash @ 149F 60min; boil 60 min; Hallertau 1.75oz @ 60 and .25oz @ 10. Target 1.045/1.011. Ferm @ 60F until complete, then lager for 2-4 weeks.

Answer 1

Yes, it can be fixed and the plan sounds fine. Make a log of this, and review when brewing in future so you can fix the process and avoid these problems in future brews.

Kölsch is lightly hopped so don't worry about adding additional hops. Your lower gravity will have increased utilization slightly so adding non-hopped wort will go some way to balance this out.

The math is fine. With the 1 gallon of 1.076 wort I get a combined gravity of 1.045.

(4.5 x 38 + 1 x 76)/5.5 = 45.

I would have pitched the yeast as soon as the 4.5 gallons were at pitching temperature to get the yeast going and thwart any other microbes in there. However, since it's being stored cold, you can wait until your 1 gallon addition is ready, since you're doing that within 24 hours.

Are you over thinking this? No, I don't think so. Although maybe doing a mini AG brew for just 1 gallon is a lot of effort. I'd probably just boil up some of the lightest DME you can get and make the 1.076 wort rather than go the AG route.